The Hénon family at the first bifurcation: a thermodynamical study I

Details The Hénon family at the first bifurcation: a thermodynamical study I The Hénon family at the first bifurcation: a thermodynamical study I

2011-10-04

arxiv.org/abs/1110.0601v1

Mathematics

Dynamical Systems

21 pages, 5 figures

Scientific paper

We study the dynamics of strongly dissipative H\'enon maps, at the first bifurcation parameter where the uniform hyperbolicity is destroyed by the formation of tangencies inside the limit set. We prove the existence of an equilibrium measure which minimizes the free energy associated with the non continuous potential $-t\log J^u$, where $t\in\mathbb R$ is in a certain interval of the form $(-\infty,t_0)$, $t_0>0$ and $J^u$ denotes the Jacobian in the unstable direction. We also prove the occurrence of a phase transition at which multiple equilibrium measures coexist and the pressure function is not differentiable.

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